A polygon is a figure lying in a plane and made of a series of straight segments which form its sides, where each of the sides has an end common with the preceding and the following side. These common ends make the corners of the polygon.
A polyhedron is a 3D geometrical shape made of polygons named faces, whose common sides are the edges. The intersections of the edges of a polyhedron are the vertices. The term polyhedron is also used to describe a solid whose border is made of polygons, with the edges of the polyhedron named the skeleton. The border of a polyhedron is generally considered closed, as all the faces are in contact with other faces with all their sides. In the present context, the definition of a polyhedron will be extended to open borders when the combination of polygonal faces results in an open border.
As mentioned above, a definition describes a polyhedron as a solid whose border is made of polygons. However, the skeletons defined by the edges of given polyhedra can form mechanisms. More specifically, some polygons of polyhedra can be deformed such that some polyhedra are deformable while satisfying the geometric constraints of polyhedra.
If the polygonal faces of a polyhedron are rigid and the angles between the polygonal faces (dihedral angles) can change, mechanisms can be obtained. For instance, an open polyhedron consisting of two polygons linked by a common side can form a mechanism if the two polygons can move with respect to one another. Some closed polyhedra are deformable, yet the deformable closed polyhedra are rare and they exist only for concave polyhedra, while all the convex polyhedra are rigid. In the publication “Polyhedra” (Cambridge University Press, 1997), Cromwell describes deformable polyhedra, and provides some examples, such as the Steffen mechanism.
A class of toys has been developed from the concept of deformable polyhedra. The toys of this class are made of rigid polygon-shaped parts that can be assembled with other polygon-shaped parts by a rotational joint between each adjacent polygon-shaped part, the axis of the rotational joint lying on the common side of the polygon-shaped parts, i.e., at the junction of the polygon-shaped parts. The rotational DOF between adjacent polygon-shaped parts (i.e., the change in dihedral angle) enables versatile construction of 3D structures and mechanisms. The sides of the polygon-shaped parts are of equal length so that all polygon-shaped parts are compatible.
U.S. Pat. No. 4,731,041, issued to Ziegler on Mar. 15, 1988, U.S. Pat. No. 5,545,070, issued to Liu on Aug. 13, 1996, U.S. Pat. No. 5,895,306, issued to Cunningham on Apr. 20, 1999, and the Jovo™ website (www.jovo.com), each disclose various connections between rigid polygon-shaped plates. More precisely, U.S. Pat. No. 4,731,041 describes interlocking fingers permitting a hinging action. U.S. Pat. No. 5,545,070 introduces swivel connectors joining the polygon-shaped parts. U.S. Pat. No. 5,895,306 describes plastic hinges formed integrally with the polygon-shaped parts. The systems Polydron and Frameworks (www.polydron.co.uk), disclose hinged rigid polygon-shaped parts and polygon-shaped frames, respectively. U.S. Pat. No. 5,472,365, issued to Engel on Dec. 5, 1995, and Geofix™ (www.geoaustralia.com), disclose rigid polygon-shaped frames to be hinged to one another. In all of the above-cited references, the geometry of each of the polygon-shaped parts cannot be modified, as the polygon-shaped parts are rigid. Pieces of the above-cited references are sold in kits comprising numerous parts representing the-various basic polygons, such as the triangle, the rectangle, the pentagon, etc.
Another concept discussed in the publication “Polyhedra” is the rigidity of the skeleton of polyhedra. It is known that the triangle is the only polygon that cannot be deformed. All the other polygons are deformable in a plane, such as the rectangle that can be deformed to a parallelogram, and the square that can be deformed to a diamond. The skeleton of a cube, formed of six squares, is flexible such that any of the faces can be deformed to a diamond, and the cube is deformed to a more general parallelepiped. The skeleton of a tetrahedron, formed of four triangles, cannot be deformed. Therefore, there are some mechanisms and some structures amongst the skeletons of convex polyhedra, if proper DOF are provided.
If all edges of the skeleton of a polyhedron can change their length simultaneously while the vertex angles are constant, the polyhedron keeps its general shape but changes its size. This type of mechanism is presented in “Regular Polyhedral Linkages” by Wohlhart (CK 2001, May 20–22, 2001, Seoul, Korea, pp. 239–244), where each face of the polyhedron includes a mechanism allowing its expansion. The mechanism obtained has one degree of freedom (DOF). Such one-DOF expansion is found in many deployable mechanisms. For example, the mechanisms of Hoberman, as disclosed in U.S. Pat. No. 4,942,700, issued on Jul. 24, 1990, and U.S. Pat. No. 5,024,031, issued on Jun. 18, 1991, describe one-DOF expansion spheres and construction members for forming such mechanisms.
If all angles of a polygon or a polyhedron skeleton can vary, the figures obtained will generally be very mobile. For instance, a four-sided polygon (e.g., a rectangle) allowing all angles thereof to change, will not remain planar. A practical example of this is given by Roger's Connection system (www.rogersconnection.com), which combines rods magnetized at their ends and steel balls in order to allow the assembly of many rods on a same ball, thus creating three-DOF spherical joints between the rods. Accordingly, Roger's Connection system can be used to form an infinite number of polygons and skeletons of polyhedra, with the balls positioned at the vertices and the rods representing the edges. The polyhedron skeletons formed by Roger's Connection system are generally deformable, with angles between the sides of the polygons constituting the faces of the polyhedra changing in a plane of the polygons, but are also deformable by losing the planarity of these polygons, due to the numerous DOF provided at the vertices by the steel balls. Structures can however be obtained if triangles are used, the latter being undeformable faces. Other systems using a similar concept include Geomag (www.constructiontoys.com), Magz (www.naturetapestry.com/magz.html), and Polygonzo™, Cuboctaflex™, Dodecaflex™, and Icosaflex™ (all at www.orbfactory.com).
As mentioned above, the possibility of assembling the sides of rigid faces by rotational joints allows the fabrication of structures, but rarely of mechanisms if they represent closed convex polyhedra (i.e., the skeletons are limited to being rigid). The rotational joints allow varying of the angle between two polygonal faces, whereby many different polyhedra can be constructed with a limited number of parts. However, for the toys using rigid faces, the possible polyhedra are limited to the available parts of the toy, as the polygon-shaped parts provided are often only the triangle, square, pentagon and hexagon. Therefore, a polyhedron having octagons, such as the truncated cube or the great rhombcuboctahedron, cannot be reproduced with the above-described rigid-face toys.
On the other hand, the possibility of varying all the angles results in mechanisms with too many DOF that do not preserve the planarity of the polygons, and hence do not preserve the polyhedral geometry. There is an exception if the parts are assembled using triangles. In this case only, it is possible to obtain structures, but rarely mechanisms with relatively few DOF.
A compromise between these two options is to allow the variation of angles in the planes of the polygons in addition to allowing the variation of the dihedral angle, while preserving the planarity of the polygons. Another level of flexibility could also be provided by allowing a variation in the length of the sides.